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Charging interest is something quite normal in the business world; there's nothing immoral about it per se. However, you wouldn't charge interest on a loan to your brother or sister, because you relate to them as family rather than as business associates. The Torah expects us to treat every Jew like a sibling (which, spiritually, all Jews are - see Tanya, ch. 32). By contrast, our relationships with non-Jews can be businesslike, so long as we treat them fairly (in keeping with their dignity as human beings). (Ramban's commentary to Deut. 23:20)

It's a matter of reciprocity. A non-Jew is allowed to charge a Jew interest, so it is only fair that a Jew should be able to do the same to the non-Jew. (Malbim's commentary, ibid.)

Malbim's answer is a little circular, but I like it
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MenachemApr 19 '12 at 17:20

@Menachem, it's not really circular if you think about it. There is nothing stopping them from charging us interest because they don't have a Torah. So we are given license to charge them as well. In other words, it's not nice to charge interest, but it's only fair, because they get to do it to us.
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Seth JFeb 28 '13 at 13:51

It is considered Lifnim Meshura hadin (beyond the letter of the law), like Hashavat aveidah (returning that which has been lost). since it is beyond the letter of the law, the obligation only applies to our brethren - the Jews.
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Yaakov PinskyFeb 28 '13 at 14:00

If they're allowed to steal from us, we can steal from them? If they can kill us (legally, but aren't actually doing so), we can kill them?
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Shmuel BrinNov 21 '13 at 23:41